Chapter 12 Exercises

Section A: Decay Law and Activity (Problems 1-8)

Problem 1 — Basic Decay Calculation

A sample contains $3.00 \times 10^{18}$ atoms of $^{137}$Cs ($t_{1/2} = 30.08$ yr). (a) Calculate the decay constant $\lambda$ in s$^{-1}$ and yr$^{-1}$. (b) Calculate the activity in Bq, MBq, and mCi. (c) How many atoms remain after 90.24 years (exactly 3 half-lives)? (d) What is the activity after 90.24 years?

Problem 2 — Specific Activity

Calculate the specific activity (in Bq/g and Ci/g) of: (a) $^{3}$H (tritium), $t_{1/2} = 12.32$ yr, $M = 3.016$ g/mol (b) $^{90}$Sr, $t_{1/2} = 28.79$ yr, $M = 89.91$ g/mol (c) $^{239}$Pu, $t_{1/2} = 2.411 \times 10^4$ yr, $M = 239.05$ g/mol (d) Rank these three isotopes by specific activity and explain physically why the ranking makes sense.

Problem 3 — Smoke Detector

A household smoke detector contains 1.0 $\mu$Ci of $^{241}$Am ($t_{1/2} = 432.2$ yr, $M = 241.06$ g/mol). (a) How many $^{241}$Am atoms are in the detector? (b) What is the mass of $^{241}$Am in the detector (in micrograms)? (c) What will the activity be after 50 years? (d) After how many years will the activity drop to 0.75 $\mu$Ci?

Problem 4 — Medical Isotope Timing

A nuclear medicine department receives a shipment of $^{131}$I ($t_{1/2} = 8.025$ d) at 6:00 AM on Monday. The shipment is certified as 50.0 mCi at the time of calibration (noon on the preceding Saturday). (a) What is the activity when the shipment arrives? (b) A patient treatment requires 15.0 mCi. On which day and at approximately what time will the source have decayed to this activity?

Problem 5 — Counting Statistics and Decay

A Geiger counter placed next to a short-lived source registers 8,000 counts per minute initially. After 3.0 hours, the rate has dropped to 1,000 counts per minute. Assume the detector efficiency and geometry remain constant. (a) What is the half-life of the source? (b) After what total elapsed time will the count rate be 100 counts per minute? (c) If the source is $^{56}$Mn ($t_{1/2} = 2.579$ h), is your answer to (a) consistent?

Problem 6 — Mean Life

(a) Show that the mean life $\tau = 1/\lambda$ by evaluating $\langle t \rangle = \int_0^\infty t \lambda e^{-\lambda t} dt$. (b) Calculate the mean life of $^{14}$C ($t_{1/2} = 5730$ yr) and $^{238}$U ($t_{1/2} = 4.468 \times 10^9$ yr). (c) Show that at time $t = \tau$, the fraction remaining is $1/e \approx 0.368$. (d) In a sample of 1000 atoms, approximately how many would you expect to survive past $t = 2\tau$?

Problem 7 — Mixing Two Activities

Two sources are placed side by side: Source A is 5.0 MBq of $^{24}$Na ($t_{1/2} = 14.96$ h) and Source B is 2.0 MBq of $^{198}$Au ($t_{1/2} = 2.695$ d). (a) Write an expression for the total activity as a function of time. (b) At what time do the two sources have equal activity? (c) What is the total activity at $t = 48$ h? (d) Sketch the total activity versus time and explain why the curve is not a simple exponential.

Problem 8 — Activation: Neutron Irradiation

A thin cobalt foil ($^{59}$Co, stable) is placed in a reactor neutron flux $\phi$ where it undergoes neutron capture: $^{59}$Co$(n,\gamma)^{60}$Co ($t_{1/2} = 5.271$ yr). The production rate is $R = N_{\text{target}} \sigma \phi$ (constant during irradiation). (a) Show that the number of $^{60}$Co atoms during irradiation satisfies $dN/dt = R - \lambda N$. (b) Solve this equation with $N(0) = 0$ to obtain $N(t) = (R/\lambda)(1 - e^{-\lambda t})$. (c) What is the saturation activity $A_{\text{sat}} = R$? (d) How many half-lives of irradiation are needed to reach 90% of saturation activity?

Section B: Decay Chains and Equilibrium (Problems 9-16)

Problem 9 — Two-Member Chain

$^{140}$Ba ($t_{1/2} = 12.75$ d) decays to $^{140}$La ($t_{1/2} = 1.679$ d), which decays to stable $^{140}$Ce. (a) Starting with pure $^{140}$Ba at $t = 0$, write the expression for $N_{\text{La}}(t)$. (b) At what time does the $^{140}$La activity reach its maximum? (c) What is the ratio $A_{\text{La}}/A_{\text{Ba}}$ at the maximum? (d) What type of equilibrium (secular, transient, or none) does this system approach?

Problem 10 — Secular Equilibrium

In a very old, sealed uranium ore sample, secular equilibrium exists throughout the $^{238}$U chain. (a) If the ore contains 1.00 kg of $^{238}$U, calculate the activity of $^{238}$U (in Bq). (b) In secular equilibrium, what is the activity of $^{226}$Ra in this sample? (c) Using the specific activity of $^{226}$Ra (from Section 12.3), calculate the mass of $^{226}$Ra in the sample. (d) What is the mass of $^{222}$Rn present at any instant? ($t_{1/2} = 3.823$ d)

Problem 11 — Milking the Moly Cow

A $^{99}$Mo/$^{99\text{m}}$Tc generator is freshly prepared with 10.0 GBq of $^{99}$Mo ($t_{1/2} = 65.94$ h) and zero $^{99\text{m}}$Tc. (a) Write the expression for the $^{99\text{m}}$Tc activity as a function of time ($t_{1/2} = 6.006$ h). (b) At what time does the $^{99\text{m}}$Tc activity reach its maximum? (c) What is the maximum $^{99\text{m}}$Tc activity? (d) If the generator is "milked" (all $^{99\text{m}}$Tc removed) at $t = 24$ h, how long must one wait for the $^{99\text{m}}$Tc to regrow to 90% of its new maximum?

Problem 12 — Branching Decay

$^{64}$Cu ($t_{1/2} = 12.70$ h) undergoes three decay modes: - $\beta^+$ (17.6%) to $^{64}$Ni - $\beta^-$ (38.5%) to $^{64}$Zn - Electron capture (43.9%) to $^{64}$Ni

(a) Calculate the partial decay constant for each mode. (b) Calculate the partial half-life for each mode. (c) A sample initially has $10^{15}$ atoms of $^{64}$Cu. How many atoms of $^{64}$Zn are produced in the first 12.70 hours? (d) How many positrons are emitted in total (from $t = 0$ to $t = \infty$)?

Problem 13 — Three-Member Chain

Consider the chain $^{238}$U $\xrightarrow{\alpha}$ $^{234}$Th $\xrightarrow{\beta^-}$ $^{234}$Pa ($\xrightarrow{\beta^-}$ $^{234}$U). Data: $t_{1/2}(^{238}\text{U}) = 4.468 \times 10^9$ yr, $t_{1/2}(^{234}\text{Th}) = 24.10$ d, $t_{1/2}(^{234}\text{Pa}) = 6.70$ h. (a) Starting with pure $^{238}$U, use the Bateman equations to write $N_{\text{Pa}}(t)$. (b) Argue that secular equilibrium is rapidly established (within weeks) for both daughters. (c) In a sample with $A(^{238}\text{U}) = 10^4$ Bq, what are the activities of $^{234}$Th and $^{234}$Pa at secular equilibrium? (d) Calculate the mass of $^{234}$Th in secular equilibrium with 1.00 kg of $^{238}$U.

Problem 14 — No Equilibrium

$^{211}$Pb ($t_{1/2} = 36.1$ min) $\to$ $^{211}$Bi ($t_{1/2} = 2.14$ min) $\to$ $^{207}$Tl (stable, via $\alpha$ to first approx.). (a) Starting with pure $^{211}$Pb, write $A_{\text{Bi}}(t)$. (b) At what time does $A_{\text{Bi}}$ reach its maximum? (c) Explain why no equilibrium condition is reached. (d) Sketch $A_{\text{Pb}}(t)$ and $A_{\text{Bi}}(t)$ on the same axes for $0 \leq t \leq 200$ min.

Problem 15 — Radon Ingrowth

A well-ventilated room suddenly has its ventilation shut off. Radon ($^{222}$Rn, $t_{1/2} = 3.823$ d) seeps in from the soil at a constant rate $R$ (atoms per second). (a) Show that the radon activity in the room approaches $A_{\text{eq}} = R$ (i.e., $A = R(1 - e^{-\lambda t})$). (b) If the equilibrium radon concentration corresponds to 200 Bq/m$^3$ in a 150 m$^3$ house, what is the source rate $R$? (c) How long after the ventilation is shut off does the radon reach 50% of its equilibrium value? 90%? (d) What mass of radon is present at equilibrium?

Problem 16 — Decay Chain Conservation

For any decay chain $N_1 \to N_2 \to \cdots \to N_n$ (stable), show that: $$N_1(t) + N_2(t) + \cdots + N_n(t) = N_{1,0}$$ That is, atoms are conserved — they merely change identity. Prove this directly from the differential equations.

Section C: Radioactive Dating (Problems 17-22)

Problem 17 — Carbon-14 Age

A charcoal sample from a campfire yields a $^{14}$C activity of 9.5 dpm/g C. The modern standard is 15.3 dpm/g C. (a) Calculate the uncalibrated radiocarbon age. (b) Express the result in the conventional "years before present (BP)" notation (where "present" = 1950 CE). (c) How much uncertainty does a $\pm 0.3$ dpm/g measurement error introduce in the age?

Problem 18 — Limits of Carbon Dating

(a) An AMS (accelerator mass spectrometry) system can measure $^{14}$C/$^{12}$C ratios as low as $10^{-15}$. Given that the modern ratio is $1.2 \times 10^{-12}$, what is the maximum age that can be measured? (b) How many half-lives does this represent? (c) Why doesn't $^{14}$C dating work for samples older than about 50,000 years, even with AMS?

Problem 19 — K-Ar Dating

A volcanic rock contains 5.0 ppm of potassium by mass and has a $^{40}$Ar/$^{40}$K ratio of 0.045 (radiogenic argon only). The natural abundance of $^{40}$K is 0.0117%. The total half-life of $^{40}$K is $1.248 \times 10^9$ yr, with the EC/Ar branch fraction 0.1072. (a) Calculate the age of the rock. (b) What fraction of the original $^{40}$K has decayed?

Problem 20 — U-Pb Concordia

A zircon crystal has the following measured ratios: - $^{206}$Pb/$^{238}$U = 0.179 - $^{207}$Pb/$^{235}$U = 1.87

(a) Calculate the age from the $^{206}$Pb/$^{238}$U ratio ($\lambda_{238} = 1.551 \times 10^{-10}$ yr$^{-1}$). (b) Calculate the age from the $^{207}$Pb/$^{235}$U ratio ($\lambda_{235} = 9.849 \times 10^{-10}$ yr$^{-1}$). (c) Are the two ages concordant? What does this imply about the zircon?

Problem 21 — Rb-Sr Isochron

Four minerals from a meteorite yield the following data:

(a) Plot these data and determine the slope of the isochron. (b) Using $\lambda = 1.42 \times 10^{-11}$ yr$^{-1}$, calculate the age. (c) What is the initial $^{87}$Sr/$^{86}$Sr ratio? Is it consistent with the solar system initial value (BABI $\approx 0.69898$)?

Problem 22 — Multi-Method Consistency

A rock is dated using three methods: - $^{238}$U-$^{206}$Pb gives $4.53 \pm 0.05$ Gyr - $^{87}$Rb-$^{87}$Sr gives $4.50 \pm 0.10$ Gyr - $^{147}$Sm-$^{143}$Nd gives $4.56 \pm 0.15$ Gyr ($\lambda = 6.54 \times 10^{-12}$ yr$^{-1}$)

(a) Are these ages concordant within uncertainties? (b) What does concordance (or discordance) tell you about the sample's history? (c) Propose a weighted average age, weighting each measurement by $1/\sigma^2$.

Section D: Natural Series and Advanced Problems (Problems 23-27)

Problem 23 — Uranium Series Chain

In the $^{238}$U decay chain: (a) Verify that 8 $\alpha$ decays and 6 $\beta^-$ decays are needed to go from $^{238}$U to $^{206}$Pb by tracking $(Z, A)$. (b) Calculate the total energy released in the complete chain, given that $M(^{238}\text{U}) = 238.05079$ u and $M(^{206}\text{Pb}) = 205.97447$ u, $M(^4\text{He}) = 4.00260$ u (ignore electron masses and neutrino energies for an estimate). (c) What fraction of this energy is carried by the $\alpha$ particles?

Problem 24 — Radon Activity in Ore

A uranium mine contains ore with a grade of 0.3% U$_3$O$_8$ by mass. The ore density is 2.7 g/cm$^3$. (a) Calculate the $^{238}$U activity per kg of ore. (b) In secular equilibrium, what is the $^{222}$Rn activity per kg? (c) If radon escapes from the ore with an emanation coefficient of 20%, what is the radon emission rate per m$^3$ of ore? (d) Why is radon the primary radiation hazard in uranium mines?

Problem 25 — Age of the Solar System from $^{237}$Np

$^{237}$Np has $t_{1/2} = 2.144 \times 10^6$ yr. If the solar system formed 4.568 Gyr ago: (a) How many half-lives have elapsed? (b) What fraction of the original $^{237}$Np remains? (c) If the solar system initially had equal abundances (by number) of $^{237}$Np and $^{238}$U, what would the $^{237}$Np/$^{238}$U ratio be today? (d) Explain why the neptunium series is called "extinct" while the other three are not.

Problem 26 — Isotope Production in a Reactor

$^{99}$Mo is produced by neutron-induced fission of $^{235}$U in a reactor. The fission yield of $^{99}$Mo is 6.1%. (a) If a reactor produces fissions at a rate of $10^{18}$ fissions/s, what is the $^{99}$Mo production rate? (b) At saturation (production = decay), what is the $^{99}$Mo activity? (c) The $^{99}$Mo is extracted weekly. What fraction of the saturation activity is reached in one week? (d) The world requires about $3.7 \times 10^{14}$ Bq (10,000 Ci) of $^{99}$Mo per week for medical imaging. How does this compare to your answer?

Problem 27 — Computational: Decay Chain Simulation

This problem accompanies the progressive project (decay_chains.py).

Using a numerical ODE solver (e.g., scipy.integrate.odeint): (a) Simulate the three-member chain $^{140}$Ba → $^{140}$La → $^{140}$Ce (stable) for $t = 0$ to $80$ days. Plot $N_1(t)$, $N_2(t)$, $N_3(t)$ and verify $N_1 + N_2 + N_3 = N_{1,0}$. (b) Simulate the approach to secular equilibrium for $^{226}$Ra → $^{222}$Rn. Start with pure Ra and plot $A_{\text{Rn}}/A_{\text{Ra}}$ vs. time in units of Rn half-lives. (c) Extend to the full $^{238}$U decay chain (14 members). Starting from pure $^{238}$U, demonstrate that after a long time all activities converge (secular equilibrium). (d) Introduce a perturbation: at $t = 10^6$ yr, remove all $^{222}$Rn from the system. Show how the chain re-equilibrates below $^{222}$Rn.